Finding derivative of x^x using limit definition

Click For Summary
SUMMARY

The discussion focuses on finding the derivative of the function x^x using the limit definition. The key limit expression discussed is lim [(x+h)^h - 1]/h as h approaches 0, which equals ln(x). Participants emphasize the importance of the exponential and logarithmic relationships, specifically using e^{ln(x)} = x and the Taylor series expansion for log(1+h) to derive the result. The conversation highlights the application of these mathematical concepts to successfully compute the derivative.

PREREQUISITES
  • Understanding of limit definitions in calculus
  • Familiarity with exponential functions and logarithms
  • Knowledge of Taylor series expansions
  • Basic proficiency in mathematical notation and manipulation
NEXT STEPS
  • Study the derivation of exponential functions using limits
  • Explore Taylor series and their applications in calculus
  • Learn about the properties of logarithms, particularly natural logarithms
  • Investigate advanced techniques in differentiation, such as implicit differentiation
USEFUL FOR

Students studying calculus, mathematicians interested in derivatives, and educators teaching advanced calculus concepts will benefit from this discussion.

scottshannon
Messages
46
Reaction score
0
I am trying to find the derivative of x^x using the limit definition and am unable to follow what I have read. Can someone help me understand why lim [(x+h)^h -1]/h as h ---> 0 = ln(x). This part of the derivatio
 

Attachments

  • 11143198_952395444799534_6323303759743605136_n.jpg
    11143198_952395444799534_6323303759743605136_n.jpg
    27.2 KB · Views: 5,444
Physics news on Phys.org
Try using the fact that e^{ln(x)}=x
 
Limit (a∧h) = 1 + h×log(a)/1! + ((h×log(a))∧2)/2! + ...
h→0

For the steps performed on the right side , use , as already said , x = e∧log(x) ,
and

Limit (log(1+h)) = h - (h∧2)/2 + (h∧3)/3 - ...
h→0

*Log is taken to base e .
 

Similar threads

High School Why does the Limit Process give an Exact Slope?
  • · Replies 53 ·
2
Replies
53
Views
7K
Undergrad Differentiability of a Multivariable function
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Explicit logical justification for last step in epsilon/delta proof?
  • · Replies 20 ·
Replies
20
Views
3K
Undergrad Uniform convergence of family of functions with continuous index
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad From a proof on directional derivatives
  • · Replies 9 ·
Replies
9
Views
2K
Undergrad Why prove Taylor's theorem with p(x)=q(x)−((b−a)^n/(b−x)^n)q(a)?
  • · Replies 9 ·
Replies
9
Views
3K
High School Proof of Chain Rule: Understanding the Limits
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad What version of the definition of derivative
  • · Replies 1 ·
Replies
1
Views
1K
MHB Derivative using the limit definition (without using L'Hospital's rule)
  • · Replies 4 ·
Replies
4
Views
2K
MHB Check existence of limit with definition
  • · Replies 2 ·
Replies
2
Views
2K